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Tag: Phase-Shift

In This earlier posting of mine, I had written about certain concepts in surround-sound, which were based on Pro Logic and the analog days. But I had gone on to write, that in the case of the AC3 or the AAC audio CODEC, the actual surround channel could be encoded separately, from the stereo. The purpose in doing so would have been, that if decoded on the appropriate hardware, the surround channel could be sent directly to the rear speakers – thus giving 6-channel output.

While writing what I just linked to above, I had not yet realized, that either channel of the compressed stream, could contain phase information conserved. This had caused me some confusion. Now that I realize, that the phase information could be correct, and not based on the sampling windows themselves, a conclusion comes to mind:

Such a separate, compressed surround-channel, would already be 90⁰ phase-shifted with respect to the panned stereo. And what this means could be, that if the software recognizes that only 2 output channels are to be decoded, the CODEC might just mix the surround channel directly into the stereo. The resulting stereo would then also be prepped, for Pro Logic decoding.

One of the open-source applications which can be used as a Sound-Editor, is named ‘Audacity’. And in an earlier posting, I had written that this application may apply certain effects, which first involve performing a Fourier Transform of some sort on sampling-windows, which then manipulate the frequency-coefficients, and which then invert the Fourier Transform, to result in time-domain sound samples again.

On closer inspection of Audacity, I’ve recently come to realize that its programmers have avoided going that route, as often as possible. They may have designed effects which sound more natural as a result, but which follow how traditional analog methods used to process sound.

In some places, this has actually led to criticism of Audacity, let’s say because the users have discovered, that a low-pass or a high-pass filter would not maintain phase-constancy. But in traditional audio work, low-pass or high-pass filters always used to introduce phase-shifts. Audacity simply brings this into the digital realm.

I just seem to be remembering certain other sound editors, that used the Fourier Transforms extensively.

A concept that exists in radio-communications, which is derived from amplitude-modulation, and which is further derived from balanced modulation, is single-sideband modulation. And even back in the 1970s, this concept existed. Its earliest implementations required that a low-frequency signal be passed to a balanced modulator, which in turn would have the effect of producing an upper sideband (the USB) as well as an inverted lower sideband (the LSB), but zero carrier-energy. Next, the brute-force approach to achieving SSB entailed, using a radio-frequency filter to separate either the USB or the LSB.

The mere encumbrance of such high-frequency filters, especially if this method is to be used at RF frequencies higher than the frequencies, of the old ‘CB Radio’ sets, sent Engineers looking for a better approach to obtaining SSB modulation and demodulation.

And one approach that existed since the onset of SSB, was actually to operate two balanced modulators, in a scheme where one balanced modulator would modulate the original LF signal. The second balanced modulator would be fed an LF signal which had been phase-delayed 90⁰, as well as a carrier, which had either been given a +90⁰ or a -90⁰ phase-shift, with respect to whatever the first balanced modulator was being fed.

The concept that was being exploited here, is that in the USB, where the frequencies add, the phase-shifts also add, while in the LSB, where the frequencies subtract, the phase-shifts also subtract. Thus, when the outputs of the two modulators were mixed, one side-band would be in-phase, while the other would be 180⁰ out-of-phase. If the carrier had been given a +90⁰ phase-shift, then the LSB would end up 180⁰ out-of-phase – and cancel, while if the carrier had been given a -90⁰ phase-shift, the USB would end up 180⁰ out-of-phase – and cancel.

This idea hinges on one ability: To phase-shift an audio-frequency signal, spanning several octaves, so that a uniform phase-shift results, but also so that the amplitude of the derived signal be consistent over the required frequency-band. The audio signal could be filtered to reduce the number of octaves that need to be phase-shifted, but then it would need to be filtered to achieve a constrained frequency-range, before being used twice.

And so a question can arise, as to how this was achieved historically, given analog filters.

My best guess would be, that a stage which was used, involved a high-pass and a low-pass filter that acted in parallel, and which would have the same corner-frequency, the outputs of which were subtracted – with the high-pass filter negative, for -90⁰ . At the corner-frequency, the phase-shifts would have been +/- 45⁰. This stage would achieve approximately uniform amplitude-response, as well as achieving its ideal phase-shift of -90⁰ at the one center-frequency. However, this would also imply that the stage reaches -180⁰ (full inversion) at higher frequencies, because there, the high-pass component that takes over, is still being subtracted !

( … ? … )

What can in fact be done, is that a multi-band signal can be fed to a bank of 2nd-order band-pass filters, spaced 1 octave apart. The fact that the original signal can be reconstructed from their output, derives partially from the fact that at one center-frequency, an attenuated version is also passed through one-filter-up, with a phase-shift of +90⁰ , and a matching attenuated version of that signal also passed through one-filter-down, with a phase-shift of -90⁰. This means that the two vestigial signals that pass through the adjacent filters are at +/- 180⁰ with respect to each other, and cancel out, at the present center-frequency.

If the output from each band-pass filter was phase-shifted, this would need to take place in a way not frequency-dependent. And so it might seem to make sense to put an integrator at the output of each bp-filter, the time-constant of which is to achieve unit gain, that the center-frequency of that band. But what I also know, is that doing so will deform the actual frequency-response of the amplitudes, coming from the one band. What I do not know, is whether this blends well with the other bands.

If this was even to produce a semi-uniform -45⁰ shift, then the next thing to do, would be to subtract the original input-signal from the combined output.

(Edit 11/30/2017 :

It’s important to note, that the type of filter I’m contemplating does not fully achieve a phase-shift of +/- 90⁰ , at +/- 1 octave. This is just a simplification which I use to help me understand filters. According to my most recent calculation, this type only achieves a phase-shift of +/- 74⁰ , when the signal is +/- 1 octave from its center-frequency. )

Now, my main thought recently has been, if and how this problem could be solved digitally. The application could still exist, that many SSB signals are to be packed into some very high, microwave frequency-band, and that the type of filter which will not work, would be a filter that separates one audible-frequency sideband, out of the range of such high frequencies.

And as my earlier posting might suggest, the main problem I’d see, is that the discretized versions of the low-pass and high-pass filters that are available to digital technology in real-time, become unpredictable both in their frequency-response, and in their phase-shifts, close to the Nyquist Frequency. And hypothetically, the only solution that I could see to that problem would be, that the audio-frequency band would need to be oversampled first, at least 2x, so that the discretized filters become well-behaved enough, to be used in such a context. Then, the corner-frequencies of each, will actually be at 1/2 Nyquist Frequency and lower, where their behavior will start to become acceptable.

The reality of modern technology could well be such, that the need for this technique no longer exists. For example, a Quadrature Mirror Filter could be used instead, to achieve a number of side-bands that is a power of two, the sense with which each side-band would either be inverted or not inverted could be made arbitrary, and instead of achieving 2^n sub-bands at once, the QMF could just as easily be optimized, to target one specific sub-band at a time.

I have run into people, who believe that a signal cannot be phase-advanced in real-time, only phase-delayed. And as far as I can tell, this idea stems from the misconception, that in order for a signal to be given a phase-advance, some form of prediction would be needed. The fact that this is not true can best be visualized, when we take an analog signal, and derive another signal from it, which would be the short-term derivative of the first signal. ( :1 ) Because the derivative would be most-positive at points in its waveform where the input had the most-positive slope, and zero where the input was at its peak, we would already have derived a sine-wave for example, that will be phase-advanced 90⁰ with respect to an input sine-wave.

But the main reason this is not done, is the fact that a short-term derivative also acts as a high-pass filter, which progressively doubles in output amplitude, for every octave of frequencies.

What can be done in the analog domain however, is that a signal can be phase-delayed 90⁰, and the frequency-response kept uniform, and then simply inverted. The phase-diagram of each of the signal’s frequency-components will then show, the entire signal has been phase-advanced 90⁰.